The synthesis of materials that superconduct at room temperature and atmospheric pressure has been a long-standing dream since the discovery of superconductivity (1). The theoretical predictions of high Tr superconductivity in metallic hydrogen (2,3) and its subsequent prediction in dense hydrogen-rich materials (4) have been realized experimentally (5-9) (see ref. 10 for a review), alongside theoretical work. Notable in this regard are predictions of extremely high-Tr superconductivity in LaHio (11,12) which were found to agree closely with experiment (5,6), followed by later experiments on the YH" system (7)(8)(9) and ternary hydrides (13)(14)(15). Several more recent studies have proposed ternary or more complex metal hydrides as vehicles to lower the pressure needed for very high-temperature superconductivity (16)(17)(18)(19)(20)(21)(22)(23)(24).

Many such studies start with the highest Tf stoichiometric compounds and examine alloying effects that lower the pressure necessary to stabilize their structures. An alternative approach is to begin with hydrogen-rich MH3 units that are already stable at ambient conditions and embed them in supercells modified to maximize Tf, as alluded to in ref. 4 and explored in a recent computational study of [Al,Ga]Hg alloys (23). The Fm3m rareearth trihydrides (RH3; R = Sc,Y,...) have strong electron-phonon coupling (EPC) (25)(26)(27)(28)(29) associated primarily with octahedrally coordinated hydrogen, as in palladium hydrides (30). Notably, experiments on YH3 indicate that the face-centered cubic (FCC) phase can be kinetically stabilized with doping (31,32), ball milling (33), and quenching (34). Doping the rare earth (35) and substituting nitrogen for octahedral hydrogen in LUH3 (36,37) have both been predicted to increase this electron-phonon coupling. Evidence of superconductivity at near-ambient conditions has been reported for material containing FCC-based LuH3_xNy; while the original work is controversial (38), subsequent studies confirmed some of the experimental results (39) and the latter work is ongoing (40). Other experiments indicate that an FCC-based phase of N-doped LuH3_3 can be recovered to ambient conditions (41) and an FCC Lu-H-N phase hypothesized to be N-doped LuH3_3 displays nearly flat electronic bands just under the Fermi energy Ey (42).

Superconductivity at ambient pressures and temperatures above the freezing point of water would enable numerous technological advances that include lossless electron transport, more cost-effective high field magnets, and magnetic levitation devices. We propose three model structures for which a variety of specific combinations of rare earth, alkaline earth, and alkali metals together with hydrogen atoms conspire to form metallic hydrogen sublattices that enable high-temperature superconductivity. These systematics lead to predicted superconducting critical temperatures above 200 K for several systems, including phases plausibly stable at modest pressures. Nearly flat bands were first predicted in Fm3m LugbNyN (43) which is predicted to be dynamically stable at 2 GPa when including nuclear quantum effects (44).

Inspired by the above work, we generalize the idea of doping metal atoms in RHy to include small amounts of doping of the octahedral hydrogen with Lewis bases [0,N,S,P]. As the space of possible quaternary structures and stoichiometries is huge, we use FCC RFiy and the structural motif from our previous work (43) to design model structures with relatively few degrees of freedom. Specifically, we propose three model structures of increasing complexity that are engineered to have dominant hydrogen states at the Ep that emulate a metallic hydrogen sublattice (4) (Fig. 1). We use these structures as starting points and perform density functional theory (DFT) variable-cell relaxations for specific compounds. We estimate Tf using a linear correlation observed between Eliashberg Tf and a quantity 5> based on electronic properties of the hydrogen sublattice in hydrides (.Materials and Methods). We also perform frozen phonon calculations near ambient pressures to examine the dynamical stability of selected structures and assess the effect of anharmonicity.

Pm3m RH11X3Y Structure. We present first a simple model structure for quaternary metal hydrides, with the chemical formula RH11X3Y (Fig. 2). It is distinguished by a central RHn unit with a typical electron localization function (ELF; see Eq. 6) and projected density of states (PDOS) illustrated using LuHn (i.e. not filling X,Y positions) as an example in Fig. 2 C and D. Relaxation was not performed as our goal is to compare the electronic structure of RFin with that of RH11X3Y with as few variables changed as possible. In this model structure, the RHn unit is surrounded by an X3Y "cage" where the hydrogens are placed in the positions they occupy in FCC RH3, except for the la Wyckoff position which may be occupied by hydrogen or a Lewis base (N,P,0,S). The 3d Wyckoff positions are also kept unspecified, but the intent for these atoms is to stabilize the FCC lattice. Choices of X and Y are also intended to make the X3Y cage inert such the chemical bonding of the central RHn unit (and therefore its superconducting properties) is preserved. We explore the effects of filling the X and Y positions and hole doping in SI Appendix, Fig. SI, finding that the addition of trivalent X atoms typically leads to strong H bonding, bringing the hydrogen states far below Ep. Hole-doping and filling Y = O both result in hydrogen states dominating the DOS(E^), though not at the same magnitude as LuHn (Fig. 2D). These structures are simple enough to perform Eliashberg calculations for Tr, unlike the more complex structures discussed later. For the latter, we estimate the critical temperature for hydride super conductors based on electronic indicators (45), which we refer to asT^.

We first note that the networking-value Tj?et (Eq. 8) was found to be within ±60 K of isotropic Eliashberg T^' calculations for 300 hydrides (45). Calculations since then have verified its accuracy at predicting Eliashberg Tfs for CaHg (T^h 236 K, T^: 264 K) (46) and L^HnM (T^: 100 K, T^: 99 K) (47). As a check on the Tf estimation process, we estimate Eq. 8 for Fm3m LuHio [SIAppendix, Fig. S2) at 150 GPa (a = 4.91 A) gives Tj?et Rti 340 K compared to the Eliashberg T^' = 290 K (A ss 3.5) for Fm3m LuHio at 175 GPa (47). Additionally, our LuHn unit in Fig. 2 (a = 5.21 A) is estimated to have a Tj?et of 265 K. Our hypothesis is that any structure with similar chemical tendencies as these FCC RHiq+x examples would have a Tr in a comparable range. We note the similarity to the Tf range reported experimentally for Lu-H-N (39). LU4H11O example. As a reference for structural, electronic, and Eliashberg calculations, we choose the ternary example Pm3m Lu4HnO (Fig. 3). The relative hydrogen orbital fraction of the density of states at Ep [DOSff(Ef)] is 0.5, meaning its predicted Tc should be higher than that of another RH11X3Y model example, Lu4HnN (47) [DOS#(E/r) ~ 0.33]. This arises from the difference in ionic bonding strength of O compared to N. We prescribe the bonding nature between rare earth atoms and hydrogen/oxygen/nitrogen atoms in the doped RH3 systems to be generally ionic using the ELF (45), with experiments on YH3 supporting this picture (48). We find through the ELF that the 0','',' = 0.521 (see the discussion below Eq. 8), with the networking value predicted T"et = 130 K. Pm3m Lu^HnO features a nearly flat electron band 0.36 eV below E/ (around X) (Fig. 30) which may contribute to the superconductivity due to the high energy of phonons around 12 to 15 THz (about 0.08 eV) and 35 THz (0.15 eV). We find increasing correlation effects via DFT+U (Uci = 5.5 on Fu) can induce Fifshitz transitions as can be observed at the F and R points of the band structure (Fig. 3C) indicating correlation effects could bring van Hove singularities (vHs) to E/7 and therefore increase the EPC as seen in H3S (49). Density functional perturbation theory calculations on a 4 x 4 x 4 q-point mesh indicate it is dynamically stable above 25 GPa. We used the Electron-Phonon Wannier (EPW) code ( 50) to interpolate the electron-phonon matrix elements using Wannier functions, with k/q meshes of 323 and 163. The a2F(m) leads to a large integrated electron-phonon coupling of A ss 3 (Fig. 3D) and a>iog % 5 THz, with an Alien-Dynes Tf (Eq. 1) of 90 K using // = 0.10 (51). The Alien-Dynes Tc decreases to 50 K if not converged (e.g. using a coarser mesh and larger thermal broadening factor). The Tf has sizable contributions from the octahedral hydrogen modes below 5 THz and 12 to 15 THz, with a small contribution to a>iog from higher-frequency tetrahedral hydrogen modes at 35 THz. We performed anisotropic Eliashberg calculations as imple mented in the EPW software (52) and compute the quasiparticle DOS in the superconducting state at various temperatures, illustrating the gap closure around 125 K (Fig. 3E) which coincides with the T"et of 130 K predicted by Eq. 8. The sizable difference between the Alien-Dynes and Eliashberg Tf reflects that the Allen-Dynes formula was derived for an EPC A < 2 (51). The phonon dispersion, superconducting properties, and effect of doping are further elucidated in {SIAppendix, Fig. S3).

Other RH11X3Y examples. Both LuH,, Ca,0 and CaH,, Y,0 have dominant DOSg(Ef) and end up having similar T(?et despite the almost threefold difference in their total DOS^^Ef) {SI Appendix, Fig. S4). We emphasize this is due to the observation that Tc is not correlated with the total DOS(Eyr) in hydrides, but the relative hydrogen contribution (45).

As boron is known to attain an oxidation state as low as B5-, we also explored LU4H11B {SIAppendix, Fig. S5) and found an interesting flat band near Eyr of Lu-d, H-s, and B-p character which remained under relaxation that would likely lead to an instability of some kind. We also found SiHnLuyN to have a hybridized nearly flat band near Ey {SI Appendix, Fig. S6) with T(?et = 90 K. As their predicted T(?et are below 200 K we did not further examine these examples. We confirm there is no DFT+U dependence for these results (the Fermi level properties do not change).

Selected examples discussed so far as well as the others we explore later are outlined in Table 1, with some relative energies of formation computed in SI Appendix, Table SI. The results establish that while the first model structure (Fig. 2) can result in hydrogen-dominant states at Ey, the magnitude of the DOS(Ey) is not comparable to that of the core RHn structure; the DOSgrf/(Ey) ratio is not typically close to 1, which limits the achievable T(?et (Eq. 8).

Fm3m (RH11)2X6YZ Structure. Using T(lvl (Eq. 8) as a guiding principle, we build on the simple RH11X3Y structure (Fig. 2) and propose a supercell to increase Hy and DOSyy(Ey). The increased Hy requires less [0,N,S,P], and we hypothesize that high symmetry Fm3m structures inspired by our previous work (43,44) will lead to large DOSh{Ef) by bringing tetrahedral hydrogen states to Ey. This leads to a supercell in which charge is redirected away from the X3Ycage surrounding the RHn unit (Fig. 4 A and B). We visualize and specify a model structure meeting these requirements in Fig. 4 C and D. The blue regions indicate the RH11 regions.

Our second model structure (Fig. 4) is designed to emulate the hydrogen-dominated band structure near Ey of the core RHn units, with the appropriate choice of X, Y, and Z atoms. The chemical formula is (RHnlyXgYZ, where R is an atom in the 8c position, X is an atom stabilizing the FCC RHn lattice in the 24e position with x % 0.25, and Y, Z in the 4a/4b positions ensure the X atoms are inert with respect to the RHn lattice. The hydrogens occupy two 32f positions with x % 0.875 and x % 0.625 as well as the 24d positions. SI Appendix, Fig. S13 shows four examples without relaxing the structure from the parameters given in Fig. AD. Their electronic properties suggest a high estimated Tc in the 150 to 170 K range using Eq. 8. However, these examples are unrelaxed so we do not quantify their properties further. (CaH11)2 Sc6NH example. We start with an in-depth look at a par ticular example of the (RHn (2X5 YZ model structure. Choosing R = Ca, X= Sc, Y= N, andZ = H, we have (CaHn)7ScgNH, or Sc0.75Ca0.25Fd2.875N0.125 • V7e focus on this example for a variety of reasons. In terms of stability, 25 to 33% Ca doping is predicted to decrease the pressure needed to stabilize cubic SCH3 by over 10 GPa (35), which we infer (based on the present work and Fig. 5 from ref. 35) is due to the reduced DOSy^(Ey) which lowers the

Table 1. Table of selected materials, with the parent structure, computed electronic properties, and estimated* Tc DOStca(Ep) Parent Structure Material a (A) P* (GPa) (states/eV/f.u.) DOS"'(EF) T?et (K) Notes RH -] 1X3Y O 4.76 25 0.36 0.66 0.50 130 T^' = 125 K with s-wave gap LuH-11 Ca30 5.11 0 0.375 0.62 0.52 145 CaH'i 1Y3O 5.14 0 0.22 0.66 0.485 130 ScH-11 Y3N 5.05 0 0.62 to 0.9* 0.5 0.495 120 SiH'i 1 LU3N 4.90 0 0.88 0.4 0.47 90 TJT* = 125 K with a supercell distortion (RHnlgXgYZ LbgH23N 10.04 1 0.43 to 1.0* 0.74 0.515 165 Dynamically stable at 2 GPa (44) LugHggN 10.07 1 0.44 0.7 0.52 160 T£et = 215 K with distortion and doping YgH%,N 10.26 1 0.5 0.55 0.5 135 Sc6Ca2H23N 9.23 20 0.375 0.5 0.582 160 T£et = 220 K with classical distortion Y6Mg2H23_xO 10.07 0 0.445 to 0.99 0.8 0.50 165 DOSt0t(Ep) doubles by removing 4b-H Mge La2H230 10.05 0 2.0 0.77 0.575 200 Dynamically unstable below 200 GPa Mg6sc2H230 9.50 0 0.67 0.77 0.575 200 Dynamically unstable below 200 GPa (R1H11)(R2Hrl)X6YZ Y7LiH23N 10.18 1 0.80 0.66 0.55 170 H sublattice may distort, remains metallic Y7LiH22N 10.22 1 0.383 0.66 0.61 190 T£et = 220 K with electron doping YyCaH23 N 10.24 1 0.572 0.65 0.56 180 jnet _ 200 K with electron doping LU7 Si H23 N 9.96 0 0.282 0.5 0.55 150 SC7 UH23N 9.33 0 0.485 0.48 0.655 190 ScyNal-l23 N 9.43 amount of electronic energy saved by the distortions. The Tf of FCC RHy was also found to increase with a critical hole doping corresponding with the enhancement of hydrogen electronic states at Eyr. This effect is reflected in the larger contributions of high-frequency modes in the Eliashberg function to A (see figure 7 in ref. 35). With respect to its potential for high-Tr superconduc tivity, CaHg has been shown to exhibit a Tr of 215 K (53) close to the original prediction (54) and possibly higher Tf with more hydrogen-rich compounds (55). Additionally, doping CaHg with rare earths could achieve even higher Tf (56,57). Finally, these ingredients satisfy the following criteria: ScN is semiconducting, the Sc atoms will help stabilize the tetrahedral hydrogen in the CaHn unit. The appreciable electonegativity ofN is expected to weaken the ionic bonding of Sc atoms with tetrahedral hydrogen due to N taking electrons from Sc that would normally form ionic bonds with hydrogen, bringing tetrahedral hydrogen states higher in energy to hybridize with the octahedral hydrogen.

The electronic and phonon properties of (CaHn^ScgNH at 20 GPa are shown in Fig. 5. The PDOS/bandstructure shows multiple hydrogen-dominant vHs near Ey, reminiscent of the flat bands in Lug FT 3 N (43,44). We find = 0.585 and T(?et = 165 K. Removing the 4b-H or replacing the nitrogen with oxygen both result in T(?et rising to 180 K {SIAppendix, Fig. S16). The phonon spectrum was computed at the F-pt but is fairly representative of vibrational properties due to the large 128-atom supercell size. The imaginary phonon frequencies at F indicate there may be a structural instability of the hydrogen sublattice.

To evaluate the likelihood of quantum effects stabilizing this structure against such an instability, we evaluate the anhar-monicity of the DFT potential energy surface (PES) associated with adiabatically displacing the crystal along the corresponding dynamical matrix eigenvector, which is dominated by hydrogen motions. We define the number of hydrogen strongly displaced in the mode to be all hydrogen whose displaced distance is more than Dmax/4, where Dmax is the maximum distance a hydrogen is displaced. We see the maximum energy gain at 20/30/40 GPa is 7/2/0.13 meV/hydrogen at an average hydrogen displacement of0.22/0.15/0.05 A. The double well PES becomes more harmonic with increased pressure, as we also found for LugHigN (44). SI Appendix, Fig. S14 shows examples of ground state densities computed for anharmonic potentials; potentials can be surprisingly anharmonic while still favoring a centered wavefunction. These results indicate that nuclear quantum effects stabilize the structure against this instability between 30 to 40 GPa. For pressure ranges where this distortion occurs, we find this distortion in fact could enhance T(?et up to 220 K based on our calculations at 20 GPa {SIAppendix, (45). As Hy in Eq. 10 is nearly constant in these examples, the only remaining parameter to maximize Tf is . Using this metric, we find the highest values in the (RHn)2(MggOH) systems, with a corresponding networking-value T(?et of 200 K. For R = La the high responsible for this is attributed to hydrogens tetrahedrally oriented with respect to Mg bending away from the oxygen atoms and forming a body-centered cubic hydrogen cage {SI Appendix, Fig. S21), with Fi-Fi distance 1.964 A between cube edges and 1.692 A between the vertices and the central H atom.

Such small H-H distances have also been observed in some intermetallic hydrides (58). For R = Sc, the hydrogen cubic cage does not form, but the smaller lattice constant appears to make up for it in terms of high values. Unfortunately, a stability analysis {SIAppendix, Stability) indicates ([La,Sc]Hn)2MggOH are dynamically unstable. However, we note the ternary system (LuHi 1 liLugHN was predicted to be dynamically stable at 2 GPa when considering quantum effects (44). Clearly, fine-tuning of the stoichiometry can be a determining factor in each structure's dynamical stability.

We find that structures with dominant hydrogen states at Ef can be obtained by choosing X = Sc/Y/La/Lu, R = Li/Na/Mg/Ca/Sr/Al/Ga/In, Y = H/O/S/N/P, and Z = H or vacant {SI Appendix). Swapping the role of the R and X atoms also results in hydrogen-dominant states at Ef {SI Appendix), though dynamical and thermodynamic stability is in question.

The properties of a few (RHi 1)2X5 YZ examples are summa rized in Table 1. While T(?et is generally higher for these examples the RH11X3Y model structure, we find in (CaHn)2 ScgNH that relaxation along instability pathways can in fact raise T(?et through large increases in {SIAppendix, Fig. S7). A similar vibrational analysis on (LuHi 1)2 LugN also reveals a classical distortion (44), which coupled with fine-tuning of the doping level, raises T(?et to 215 K. Based on these observations, we propose our last model structure, modified to increase and therefore T(?et. positions are summarized in Fig. 1. The change in metal ion size enhances symmetry breaking of the 24d-H positions in the structure. Those hydrogen atoms are observed to shift to either R1 or R2 (typically whichever has more valence electrons). Fig. 1 shows the symmetry-breaking hydrogens (red) as closer toward the 4d-R2 atoms.

We use this generalization of the Fm3m (RER 1)2X5 YZ to simulate 12.5% doping of ternary R8H23N structures, which involves choosing X = R2 = a rare earth, and R1 a dopant atom. Doping R8H23N. We begin with the interesting electronic structure ofLu8H23N (A) (43) and investigate replacing Lu with Sc,Y. Cell relaxation was performed and the resulting PDOS and electronic properties are compared in SI Appendix, Fig. S20. U-values were chosen by performing DFT+U calculations on the rare earth nitrides and adjusting Uj to match the indirect band gaps to experiment (59). For Y8H23N, while the flat hydrogen band is still present near E/7, the larger lattice constant results in a smaller , lowering the estimated T" '. For ScgF^N the much smaller lattice constant leads to a higher but the Sc4 orbitals hybridizing with the Fermi level leads to more charge screening of the negative nitrogen atoms. This brings hydrogen states below Ep and weakens the flatness of the hydrogen band.

Given that doping YH3 assists in stabilizing the FCC phase (31,(60)(61)(62) and various studies indicate doping rare-earth hydrides yields high T,,s (35,63,64), we investigate the effects of substituting some R = [Sc,Y] atoms in R8H23N with alkali and alkali earth atoms. In particular, we investigate Li, Na, and Mg doping in ScgE^jN, and Li, Mg, and Ca doping in Y8H23N. We start with the parent structure in Fig. 1 (32). Fig. 6 illustrates its electronic and vibrational properties via DFT and frozen phonon calculations (65). As in previous examples, the DOS//(E/-) is high and dominant (Fig. 6A), with contributions from all hydrogen, though the largest contribution is from the 4b-H. The band structure illustrates a propensity of vHs near E/7, as well as flat band regions 0.3 eV below and 1.1 eV above E/7 (Fig. 6B). As intended, the symmetry-broken octahedral hydrogen improves the connectivity of the hydrogen sublattice, increasing to 0.6 (note our record of for Y-based hydrides using the previous structure was 0.55; see SI Appendix, Fig. S10). This increases T'"1 to 185 K, which may be increased by pressure and further fine-tuning of the stoichiometry. We find electron doping can increase T'"'1 up to 220 K. Since the zero pressure phonon calculation reveals imaginary modes (Fig. 60), we perform an anharmonicity analysis as was done previously for (CaHn^ScgNH (Fig. 5). Here, we find another double well, which at 1 GPa is quite deep but becomes shallow at 5 GPa and is gone at 15 GPa (Fig. 6D). This implies the high-symmetry F43m structure becomes dynamically stable in the 5 to 15 GPa range. The distortion expected to occur below 15 GPa features octahedral hydrogen bending toward tetrahedral hydrogen, which is expected to anisotropically change . The distortion is sizable at 1 GPa yet has a fairly negligible effect on the PDOS as in (CaHn^ScgNH. The associated T'"1 drops to 145 K due to an undesirable anisotropy in 0),"'.

Various perturbations on (YH11 )(LiH11 jY^HN also seem to yield high T" '. Removing the 4b-H yields F43m Yq.875 Lio.125H2.75No.125 with T"et = 200 K and a similar double well PES for the mode with largest imaginary frequency {SIAppendix, Fig. SB). The primary difference from removing the 4b-H is the DOS is fairly flat around E/.\ but an extremely flat hydrogenbased band exists 1 eV above E/.\ corresponding to "hydrogencage-like" nearly localized electronic states. Further changing one of the 4a-N to a hydrogen atom effectively electron-dopes the system, which also bas a high T'"1 of 200 K after relaxation. 1 with PDOS plots in SI Appendix. Behavior similar to Li-doped Y8H23N occurs with H shifting toward the 4d-Sc Wyckoff positions. This change coupled with the smaller lattice constant of the FCC SCH3 parent structure leads to large due to the robust hydrogen network that forms around the 4d-Sc positions.

We summarize the properties of these examples in Table 1. The Tf estimates of 185 to 230 K for these structures illustrate the power of combining both nitrogen and alkali metal dopants in multicomponent hydrides, enabling heightened . We mention we briefly investigated replacing N with S and found F43m Sc7MgH23S is insulating but with a high 0T1 = 0.655. Hole-doping, for example with further Mg (perhaps using Fm3m (MgHi 1 )2Sc6SH) would create a metal with hydrogen-dominant states.

In the search for hydride superconductors with very high T, , a complicating factor is the necessity to include quantum zero-point motion/energy and thermal effects. Without such effects, harmonic phonons with classical nuclei often overestimate

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structural instabilities for high Tc hydrides (66,67). In fact, Tc tends to be highest when the system is closest to structural transitions (68) (and where DPT methods can fail). Another com plicating factor for first principles searches is that of metastability; it may prove sufficient for experimental purposes to produce a metastable superconductor (so long as its lifetime is measured in months or years), yet such structures would be discarded in many computational structure searches. We therefore take a different approach and propose complex hydride structures with the intent of producing electronic properties that correlate with high Tc (45). We use FCC RHy as the parent structure, as these structures have high EPC and can be stabilized at ambient pressures in a variety of ways (33-35, 41, 60).

We have proposed three successively more chemically complex model metal hydride structures with predicted Tj?et maximized by having hydrogen-dominant states at E/r and a strongly connected hydrogen sublattice. The three models exhibit increased flexibility for tuning stability and Tf relative to simple binary structures. Examples within each model structure are candidate high Tc superconductors that are potentially stable near ambient pressure and therefore deserve additional study.

Compounds based on the first and simplest model structure show moderate promise in producing ternary/quaternary hydride superconductors with Tc in the 100 to 150 K range and are potentially metastable near ambient pressures. We find Pm3m LU4H11O has a lower formation energy than LuHg + dLuhh + EhO by 0.128 eV/atom {SIAppendix, Table SI). We also note YH3 and Y2O3 are known to react to form cubic YHxOy under heating (69). The second and third model structures of quaternary hydrides have enough flexibility to optimize parameters going into the Tj?et estimator, which we use to generate examples with upper estimated Tj?ets up to 220 to 230 K at near-ambient pressures when accounting for hydrogen distortions. While the examples discussed are not on the convex hull {SIAppendix, Table SI), this does not discount their possible metastability. For example, diamond is also 0.14 eV/atom above its convex hull using data from ref. 70.

The second model can produce compounds with high Tj?et stable at low pressures, as Lug FT 3 N (A) studied by us previously (43) was shown to be dynamically stable at 2 GPa when nuclear quantum effects are included (44). Recently, N-doped FCC LUH3 has been observed to be recoverable to ambient (41) using the techniques in ref. 34. Our Fm3m (CaHn)2 ScgNH example is expected to be dynamically stable above 20 GPa when nuclear quantum effects are included. As the double well is deepened (fromi.e. lower pressure), the octahedral hydrogen are expected to distort from high symmetry positions which surprisingly increases Tj?et, with a pressure regime where spectroscopic signatures are broadened due to quantum nuclear effects. This interpretation on the structural dynamics is supported by high-pressure studies ofScHs (71).

Our third model structure involves complexity higher than quaternary systems. Nevertheless, we limit ourselves to the qua ternary case by analyzing the doping of [Sc/YjgFTaN structures. The structure is designed to distort hydrogen to increase 0T) with many possible dopants. Based on our calculations on cubic Y0_875 Li0.125 Id-2.875-<5-^0.125-c structures, we predict Tf to have a maximum of 220 K ± 60 K at a pressure between 5 to 15 GPa for optimal 8, e. For synthesis, we recommend leveraging the approaches used in refs. 32, 34, and 41. At lower pressures, distortions away from the F43m structure occur. It is difficult to predict what the effect on Tc is because the $"7 parameter becomes very anisotropic; small four-center hydrogen bond networks form for which = 0.8 but extended hydrogen networks form across the entire cell at $"7 = 0.5. Such anisotropic behavior in large unit cells was not considered in the data for Tj?et (45), so the predicted Tj?et of 145 Kfor the distorted structure at 1 GPa could have a larger error bar than 60 K. Other examples of the F43m model structure similarly illustrate that doping RgfT^N increases Tj?et by 30 to 50 K. As the highest predicted Tj?et comes from cubic (Mg,N)-doped SCH3, we note Mg and Sc form cubic alloys (72) and in fact have been studied as hydrogen storage materials (73,74). Based on our results and previous studies (31,35,60), we find it likely that these doped [Sc,Y]gFT3-3 [N.O] examples with extremely high predicted Tc are synthesizable and dynamically stable with compression.

We note again Tj?et only reproduces Tc from Eliashberg calculations within ±60 K, so some examples presented may in fact have Tc exceeding 270 K. It is likely that the Tc for the structures presented here is large in a narrow pressure range, based on Eliashberg calculations on YH3 (75) and stillcontroversial experiments on N-doped LUH3 (38,39). We remark that preliminary constrained DFT calculations indicate strong correlation effects may be present in these systems for the same reasons as presented in ref. 76; that is, DFT struggles to represent the near-neutral hydrogen atoms present in RFdg-g and how its charge fluctuations impacts the electronic structure. A more thorough investigation of these effects is underway.

In summary, we have outlined design principles for creating ternary, quaternary, and higher-order hydrides that may be high Tf or even near ambient superconductors. We presented model structures based on doped supercells of FCC RH3 to adopt their dynamical stability properties at low pressures while tuning their electronic properties. These supercells have been engineered to have high Tfs with upper estimates of 220 ± 60 K. We end by remarking that it is compelling that the highest predicted Tf using our methodology comes from (Mg,N)-doped SCFI3, which features both Ashcroft's proposal for hydrogendominant metallic alloys (4) and Feynman's guess that the highest temperature superconductor would be Sc-based (77).

We use DFT (78)(79)(80) and DFT+U (81,82) first-principles electronic structure calculations to study the examples built using our proposed model structures (Fig. 1). The Standard solid-state pseudopotentials database (83) was used to vet pseudopotentials for convergence of the computed pressure, phonon freguencies, and formation energies. Each relaxed structure had remaining forces under 1 e-5 Ry/Bohr, and energy differences were under 1 e-5 Ry. We used wavefunction plane-wave cutoffs of 75 Ry for the structures and charge density cutoffs of 300 Ry, though we performed convergence tests with higher values (especially for Sc-based compounds) and found no differences. We generally used tight k-meshes corresponding to 0.13 A-1 but did convergence tests with more k-points as needed. We used Quantum Espresso's default atomic projections for the PD0S calculations. The units for all DOS plots is 8 x states/eV/f.u. The phonopy software (65) was used to compute phonon dispersions and generate the eigenvectors for atomic displacements.

The Alien-Dynes formula for Tc takes the form [3]